Boundaries of Asplund spaces

BOUNDARIES OF ASPLUND SPACES

B. CASCALES, V. P. FONF, J. ORIHUELA, AND S. TROYANSKI

ABSTRACT. We study the relationship between the classical combinatorial inequalities ofSimons and the more recent (I)-property of Fonf and Lindenstrauss. We obtain a charac-terization of strong boundaries for Asplund spaces using the new concept of finitely selfpredictable set. Strong properties for w∗-K-analytic boundaries are established as well asa sup− lim sup theorem for Baire maps.

1. INTRODUCTION

All Banach spaces are real in this paper. Let K ⊂ X∗ be a w∗−compact convex subsetof a dual Banach space X∗. A subset B ⊂ K is called a boundary of K, if for any x ∈ Xthere is f ∈ B with maxx(K) = f(x). If K = BX∗ then a boundary B of K is called aboundary of X. From the Krein-Milman theorem follows that the set extK of all extremepoints of K is a boundary. Easy examples show that a boundary may be a proper subsetof extK. It is also possible (for a non-separable X) that a boundary is disjoint with extK.Hahn-Banach separation theorem shows that

coBw∗

= K. (W)

One of the main problems in studying boundaries is to find conditions under which aboundary B has property

coB‖·‖

= K. (S)

A boundary with (S) is also called strong. For instance, if a boundary B is separable thenit has (S) (see [30, 14, 13]). In a non-separable case this is not true: think of extBC([0,1])∗ .Although not all boundaries are strong, it has been proved in [13] that any boundary hasthe following property (I):

For any increasing sequence {Bn}∞n=1 of subsets of B such that B =∪nBn we have

K =⋃n∈N

coBnw∗‖·‖

. (I)

Property (I) is weaker than (S). However in some cases (I) implies (S). For instance (I)implies (S) for separable boundaries and for any boundary when X is separable withoutcopies of `1, see [13]. Therefore the validity of (I) for any boundary yields results byRode [30] and by Godefroy [14].

0B. Cascales, J. Orihuela and S. Troyanski were partially supported by MTM2008-05396/MTM FondosFEDER and Fundacion Seneca 08848/PI/08 CARM. V. P. Fonf was partially supported by the Spanish gov-ernment, grant MEC SAB 2005-016. S Troyanski was partially supported by Institute of Mathematics andInformatics Bulgarian Academy of Sciences, and grant DO 02-360/2008 of Bulgarian NSF

Date: April 1st. 2009 - Revised version April 7th. 2010.2000 Mathematics Subject Classification. Primary: 46B20, 46B22, 46B50,Key words and phrases. Banach spaces, Asplund spaces, James boundary.

1

2 B. CASCALES, V. P. FONF, J. ORIHUELA, AND S. TROYANSKI

A classical and important tool for the investigation of boundaries is the following Si-mons’ inequality:

For any boundary B of the w∗-compact set K in X∗ and every boundedsequence (zn) in X the following inequality holds:

supb∗∈B{lim sup

n→∞zn(b

∗)} ≥ inf{supKw : w ∈ co{zn : n ∈ N}

}. (SI)

Simons’s inequality and (I)-property look different and certainly their proofs are different.Nonetheless, Kalenda has implicitly proved, using some additional Simons’ result, that the(I)-property is equivalent to the following sup− lim sup Theorem by Simons, see [31, 32],see lemma 2.1 and remark 2.2 in [22]:

For any boundary B of the w∗-compact set K in X∗ and every boundedsequence (zn) in X the following equality holds:

supb∗∈B{lim sup

n→∞zn(b

∗)} = supx∗∈K

{lim supn→∞

zn(x∗)}. (SLS)

In Section 2 we give a proof of the fact that (I)-property, the sup− lim sup Theorem (SLS)and Simons’ inequality (SI) are indeed equivalent for any subset B of w∗-compact convexK ⊂ X∗, see Theorem 2.2; as a consequence we obtain Corollary 2.3 that, in particular,shows that neither for Simons’ inequality nor for (I)-property is important that B is aboundary (what matters is that coB‖·‖ contains a boundary). This observation could beuseful for further applications. We stress that we also prove, at the end of the paper,see Theorem 5.9, a version of the Simons’ sup− lim sup Theorem, when `1 6⊂ X , forbounded sequences (zn) in X∗∗, instead of sequences in X , but requiring that each zn is aBaire map when restricted to (BX∗ , w∗).

In the remaining of the paper we are mostly interested in boundaries of Asplund spaces.One of the main results, see Theorem 3.4, is a necessary and sufficient condition for aboundary to have (S). The main tool here is a new notion we introduce, namely, the notionof finitely-self-predictable set, FSP in short, see Definition 3.1. The definition of FSP-set isinspired by properties of boundaries of polyhedral spaces [11], and by properties of bound-aries described with σ-fragmentable maps [2, 4, 8, 17]. The discussion on σ-fragmentablemaps needs some background which is done in Section 4. As to the boundaries of polyhe-dral spaces we can do it now just to give the reader a feeling what an FSP-set is. Recall thata Banach space X is called polyhedral [24] if the unit ball of each of its finite-dimensionalsubspace is a polytope. The following theorem was proved in [11].

Theorem 1.1 ([11]). Let X be a polyhedral space of the density character w. Then X hasa boundary B ⊂ SX∗ of cardinality w such that for any h ∈ B we have

int{x∈X: h(x)=1} {x ∈ SX : h(x) = 1} 6= ∅ (1.1)

In particular the boundary B is a minimum among all boundaries, i.e., it is contained inany other boundary of X.

For a finite subset σ ⊂ X denoteEσ = spanσ. Since the unit ballBEσ of a subspaceEσis a polytope, it follows that there is a finite subset Aσ ⊂ B (Aσ may not be unique) suchthatAσ|Eσ is a boundary ofEσ (which is easily seen to be equivalent to extBE∗

σ= Aσ|Eσ ).

Thus we can define a map ξ : FX ⇒ FB from the family FX of all finite subsets of Xinto the family FB of all finite subsets of B such that ξ(σ) = Aσ. Let σn, n = 1, 2, ...,be an increasing sequence of finite subsets of X and E = [∪∞n=1σn]. By using (1.1) inTheorem 1.1 above it is not difficult to prove that the set D = ∪∞n=1ξ(σn)|E is a boundary

BOUNDARIES OF ASPLUND SPACES 3

of E. We can look at this result in the following way. We form a subspace E by usingcountably many steps (on the n-th step we add a finite-dimensional subspace Eσn), andon each step we are allowed to choose a finite subset (Aσn) of the boundary. Finally, “atthe end”, we get a boundary of E (just as the union of the sets Aσn’s): in a sense, theboundary is finitely-predictable. Clearly, this property is very strong, and it is held forpolyhedral spaces only (just take all σn’s are equal to the same finite set σ). However, asmall modification of this property (which we call FSP, see Definition 3.1) allows us toprove the following:

Theorem 3.9. Let X be an Asplund space and B be a boundary of X. Then B has (S) ifand only if B is FSP.

To prove the “if” part of Theorem 3.9 we use a separable reduction (similar to ones usedin [2, 4, 8, 12]); in the proof of the “only if” part we use the existence of the so-calledJayne-Rogers selector for the duality mapping in Asplund spaces.

We also give a characterization of Asplund spaces involving FSP boundaries, Theorem3.10, and σ−fragmented selectors, Section 4.

In Section 5 we strengthen the property (I) of boundaries (see [13]) for Asplund spaces(see Proposition 5.3), by using the γ-topology instead of the w∗-topology. By using aHaydon’s result [16] and a γ-topology technique developed in Section 5 we prove:

Theorem 5.6. Let X be a Banach space. The following statements are equivalent:(i) X does not contain `1;

(ii) for every w∗-compact subset K of X∗ any w∗-K-analytic boundary B of K isstrong.

We use standard Geometry of Banach spaces and topology notation which can be foundin [21] and [7, 23]. In particular, BE (resp. SE) is the unit ball (resp. the unit sphere) ofa normed space E. If S is a subset of E∗, then σ(E,S) denotes the topology of pointwiseconvergence on S. Given x∗ ∈ E∗ and x ∈ E, we write 〈x∗, x〉 and x∗(x) to indistinctivelydenote the evaluation of x∗ at x. If (X, ρ) is a metric space, x ∈ X and δ > 0 we denoteby Bρ(x, δ) (or B(x, δ) if no confusion arises) the open ρ-ball centered at x of radius δ.The notation B[x, δ] is reserved to denote the corresponding closed balls.

2. THE (I)-PROPERTY AND SIMONS’ INEQUALITY

Let us recall now the combinatorial principle that lies behind in James’ compactnesstheorem as it was found by S. Simons [31], and described in his famous lemma:

Lemma 2.1 (Simons, [31, Lemma 2 and Theorem 3]). Let K be a set and (zn)n a uni-formly bounded sequence in `∞(K). If B is a subset of K such that for every sequence ofpositive numbers (λn)n with

∑∞n=1 λn = 1 there exists b∗ ∈ B such that

sup{∞∑n=1

λnzn(y) : y ∈ K} =∞∑n=1

λnzn(b∗),

thensupb∗∈B{lim sup

n→∞zn(b

∗)} ≥ inf{supKw : w ∈ co{zn : n ∈ N}

}(2.1)

andsupb∗∈B{lim sup

n→∞zn(b

∗)} = supx∗∈K

{lim supn→∞

zn(x∗)}. (2.2)


Note that (SI) and (SLS) are particular cases of the thesis in Lemma 2.1 above. Ascommented in the introduction our main result in this section analyzes the coincidenceof the above (I)-property, the sup− lim sup Theorem and Simons’ inequality (2.1) in anarbitrary setting. Our proof is self-contained and uses elementary facts.

Theorem 2.2. Let X be a Banach space and K a w∗− compact and convex subset of X∗.For a given subset B ⊂ K, the following statements are equivalent:

(i) For every coveringB ⊂⋃∞n=1Dn by an increasing sequence ofw∗−closed convex

subsets Dn ⊂ K, we have

∪∞n=1Dn‖·‖

= K. (2.3)

(ii) For every bounded sequence (xk) in X

supf∈B

(lim supk

f(xk)) = supg∈K

(lim supk

g(xk)). (2.4)

(iii) For every bounded sequence (xk) in X

supf∈B

(lim supk

f(xk)) ≥ inf∑λi=1,λi≥0

(supg∈K

g(∑

λixi)). (2.5)

(iv)coB

w∗= K, (2.6)

and for every bounded sequence (xk) in X

supf∈B

(lim supk

f(xk)) ≥ inf∑λi=1,λi≥0

(supg∈B

g(∑

λixi)). (2.7)

In particular all of them happen when the subset B is a boundary of the compact K afterLemma 2.1 or (I)-property.

Proof. (i)⇒(ii). For a given bounded sequence (xk) in X let us put

l := supf∈B

(lim supk

f(xk))

and fix ε > 0. We define the sets

Dn = {h ∈ K : h(xk) ≤ l + ε, k > n}, n = 1, 2, . . .

Clearly, each Dn is w∗−closed and convex, and B ⊂⋃∞n=1Dn. Moreover, we have

lim supk f(xk) ≤ l + ε for any f ∈⋃∞n=1Dn. The assumed condition (i) implies that

the union⋃∞n=1Dn is norm-dense in K. This fact together with the boundedness of the

fixed sequence (xk) easily leads us to

supg∈K

(lim supk

g(xk)) ≤ l + ε.

Since the fixed ε > 0 is arbitrary (ii) follows.

(ii)⇒(iii) For a w∗-compact convex subset K ⊂ X∗ and c ∈ R we define

Kc = {x ∈ X : maxx(K) ≤ c}, Kcc = {f ∈ X∗ : sup f(Kc) ≤ c}.The following well-known Fact (which is an easy consequence of a separation theorem)will be used in the proof..Fact.(1) If c > 0 then Kcc = {λg : 0 ≤ λ ≤ 1, g ∈ K}.(2) If c < 0 and 0 6∈ K then Kcc = {λg : 1 ≤ λ, g ∈ K}.In all cases intKc = {x ∈ X : maxx(K) < c} 6= ∅.


Also we will use the following trivial observation. Let a functional f separates two setsA and B, i.e. inf f(A) ≥ α ≥ sup f(B) with α 6= 0. Then by passing to a multiple of fwe can get instead of α in the inequality above any real number β with αβ > 0.

PutC = co{xk}, a = inf∑

λi=1,λi≥0(supg∈K

g(∑

λixi)) = infx∈C

supg∈K

g(x).

In view of (iii) we need to prove that supf∈K lim supk f(xk) ≥ a. Assume to the contrarythat supf∈K lim supk f(xk) < a. Let b = supf∈K lim supk f(xk), if a ≤ 0, and if a > 0then let b be any number such that b ≥ supf∈K lim supk f(xk) and a > (a + b)/2 > 0.We always have

b ≥ supf∈K

lim supk

f(xk). (2.8)

Next if a ≤ 0 then b < 0, and hence 0 6∈ K. If a > 0 then from what is said above followsthat (a+ b)/2 > 0. Put c = (a+ b)/2. Clearly c 6= 0 in both cases. It follows from Factthat

intKc = {x ∈ X : maxx(K) < c} 6= ∅.Put

δ =a− b

2 supg∈K ||g||, C1 = cl{C + δBX}.

An easy calculation shows that

infy∈C1

maxg∈K

g(y) ≥ c.

Indeed, for any y = x+ z, x ∈ C, ||z|| ≤ δ, we have

maxg∈K

g(y) ≥ maxg∈K

g(x)− δ||g|| = maxg∈K

g(x)− a− b2 supg∈K ||g||

||g|| ≥ a− a− b2

= c.

Therefore C1 ∩ intKc = ∅, and by a separation theorem there is t1 ∈ X∗ with

inf t1(C1) ≥ c ≥ sup t1(Kc), (2.9)

where we used subsequently that 0 ∈ intC1 and the observation after Fact. From the righthand side in inequality (2.9) we obtain t1 ∈ Kcc. By Fact

t1 = λt, λ ≥ 1, t ∈ K, if c < 0,

t1 = λt, 0 ≤ λ ≤ 1, t ∈ K, if c > 0,

(recall that c 6= 0.) From the left-side inequality in (2.9) we deduce (in both cases: c < 0and c > 0) that inf t(C) ≥ c. Since c > b we obtain that lim supk t(xk) ≥ c > b,contradicting (2.8). The proof is complete.

(iii)⇒(i). We shall do it by contradiction. Let us fix an increasing sequence Dn of w∗

closed and convex subsets of K such that B ⊂⋃∞n=1Dn. Let us proceed by contradiction

and assume that there is z∗0 ∈ K such that z∗0 6∈ ∪∞n=1Dn‖·‖. Fix δ > 0 such that

B[z∗0 , δ] ∩Dn = ∅, for every n ∈ N.

The separation theorem in (X∗, w∗) applied to the w∗-compact sets B[0, δ] and Dn − z∗0provides us with xn ∈ X , ‖xn‖ = 1, and αn ∈ R such that

infv∗∈B[0,δ]

xn(v∗) > αn > sup

y∗∈Dnxn(y

∗)− xn(z∗0).


We have−δ = inf

v∗∈B[0,δ]xn(v

∗),

and consequently we have produced a sequence (xn)n in BX such that for each n ∈ N wehave

xn(z∗0)− δ > xn(y

∗) for every y∗ ∈ Dn. (2.10)

Fix x∗∗ ∈ BX∗∗ a w∗-cluster point of the sequence (xn)n and let (xnk)k be a subsequenceof (xn)n such that x∗∗(z∗0) = limk xnk(z

∗0). We can and do assume that

xnk(z∗0) > x∗∗(z∗0)−

δ

2, for every k ∈ N. (2.11)

Since B ⊂ ∪nDn and (Dn)n is increasing, given b∗ ∈ B there exists k0 ∈ N such thatb∗ ∈ Dnk for every k ≥ k0. Inequality (2.10) implies now that

x∗∗(z∗0)− δ ≥ lim supk

xnk(b∗) for every b∗ ∈ B. (2.12)

On the other hand, inequality (2.11) implies that

w(z∗0) > x∗∗(z∗0)−δ

2, for every w ∈ co{xnk : k ∈ N}. (2.13)

Now (iii) applies to conclude

x∗∗(z∗0)− δ(2.12)≥ sup

b∗∈Blim sup

kxnk(b

∗)(iii)

≥ inf{supKw : w ∈ co{xnk : k ∈ N}

}≥

≥ inf{w(z∗0) : w ∈ co{xnk : k ∈ N}

} (2.13)> x∗∗(z∗0)−

δ

2,

From the inequalities above we obtain 0 ≥ δ which is a contradiction that finishes theproof.

(iii)⇒(iv). Observe that (2.6) follows from (i) (take Dn being the w∗ closed convex hullof of B, n = 1, 2, ...), and (2.7) follows trivially from (iii).

(iv)⇒(iii). From (2.6) follows that

inf∑λi=1,λi≥0

(supg∈K

g(∑

λixi)) = inf∑λi=1,λi≥0

(supg∈B

g(∑

λixi)),

and the proof is over. �

The following corollary strengthens Simons’ inequality and (I)-property of boundaries.

Corollary 2.3. Let X be a Banach space, K ⊂ X∗ be a w∗-compact and convex subsetwith a boundary B1 ⊂ K. Then any subset B ⊂ B1 with coB

‖·‖ ⊃ B1, enjoys all theproperties (i)-(iv) of Theorem 2.2.

Proof. The inclusions B ⊂ B1 ⊂ coB‖·‖ imply that for any bounded sequence (xk) in X

the following equality holds:

supf∈B

(lim supk

f(xk)) = supg∈B1

(lim supk

g(xk)).

Since B1 is a boundary of K, it follows from Simons’ Lemma 2.1 that

supf∈B1

(lim supk

f(xk)) = supg∈K

(lim supk

g(xk)).


Therefore it follows

supf∈B

(lim supk

f(xk)) = supg∈K

(lim supk

g(xk)),

i.e. B satisfies (ii) in Theorem 2.2. Hence B enjoys all the equivalent properties fromTheorem 2.2. The proof is complete. �

3. FINITELY SELF PREDICTABLE SETS

Let us denote by FA the family of finite subsets of a given set A.

Definition 3.1. Let X be a Banach space and C ⊂ X∗. We say that C is finitely-self-predictable (FSP in short) if there is a map

ξ : FX → FcoC

from the family of all finite subsets of X into the family of all finite subsets of coC suchthat for any increasing sequence σn in FX , n = 1, 2, ..., with

E =[∪∞n=1 σn

], D = ∪∞n=1ξ(σn),

we haveC|E ⊂ co(D|E)

‖·‖.

Remark 3.2. If C is separable then it is FSP. Observe also that if C is FSP then, clearly Cis FSP. So we always assume without loss of generality that C is closed.

The following proposition shows that if we allow for the sets ξ(σ) in Definition 3.1 tobe countable, we get an equivalent definition.

Proposition 3.3. If there is a map ξ1 defined onFX with ξ1(σ) countable for each σ ∈ FXand satisfying the conditions in Definition 3.1, then there exits a map ξ defined on FX withξ(σ) finite for each σ ∈ FX and satisfying the conditions in Definition 3.1.

Proof. Assume that ξ1(σ) is countable. To prove the proposition it is enough to constructa map ξ(σ) with |ξ(σ)| <∞, and such that for any increasing sequence σn in FX we have

∞⋃n=1

ξ1(σn) ⊂∞⋃n=1

ξ(σn). (3.1)

Let ξ1(σ) = {fi}∞i=1. Define Pn(ξ1(σ)

)= {fi}ni=1. Next we define ξ(σ) as follows. Let

|σ| = m < ∞, and let A be the family of all non-empty subsets of σ (including σ itself).Define

ξ(σ) =⋃ν∈A

Pm(ξ1(ν)

)(3.2)

To prove (3.1) assume that f ∈ ∪∞n=1ξ1(σn). Then there are n0 and m0 such that f ∈Pm0

(ξ1(σn0)

). Since |σn| → ∞, n→∞, it follows that there is n1 with |σn1 | > m0 and

n1 > n0. It follows from the definition (3.2) that f ∈ ξ(σn1). The proof is complete. �

Theorem 3.4. Let X be a Banach space. Assume that K ⊂ X∗ is a w∗−compact convexset with boundary B ⊂ K. If B is FSP then B has (S).

Proof. Put C = coB and assume to the contrary that there is f0 ∈ K \ C‖·‖. By theseparation theorem there is F0 ∈ SX∗∗ with

supF0(C) = supF0(C‖·‖

) = α < F0(f0). (3.3)


By Goldstein’s theorem we find x1 ∈ SX with

|(F0 − x1)(f0)| < 1.

Let us write ξ({x1}) = {h1j}p1j=1 ⊂ C and let us use Goldstein’s theorem again to findx2 ∈ SX with

|(F0 − x2)(f0)| < 1/2, |(F0 − x2)(h1j)| < 1/2, j = 1, 2, · · · , p1.Proceeding by recurrence we assume that x1, x2, . . . , xn ∈ SX have been constructed andwe let ξ({x1, x2, . . . , xn}) = {hn j}pnj=1 ⊂ C. By Goldstein’s theorem we find xn+1 ∈ SXwith

|(F0 − xn+1)(f0)| < 1/(n+ 1),

|(F0 − xn+1)(hij(i))| < 1/(n+ 1) for i = 1, 2, . . . , n and j(i) = 1, 2, . . . , pi.

If we define

E = [{xi}∞i=1], D = ∪nξ({x1, ..., xn}) = {hij(i) : i ∈ N and j(i) = 1, 2, . . . , pi}we have the following properties.

(α) For each h ∈ D there is l ∈ N such that |(F0 − xm)(h)| < 1/m for every m > l.(β) |(F0 − xm)(f0)| < 1/m for every m ∈ N.

Since B is FSP it follows that B|E ⊂ co(D|E)‖·‖, and, in particular, B|E is separable. On

the other hand, B|E is a boundary of K|E and therefore (see [30, 14, 13]) co(B|E)‖·‖

=K|E . By taking into account that D ⊂ C finally we can write

C|E‖·‖

= co(D|E)‖·‖

= K|E (3.4)Let G ∈ BE∗∗ be any w∗−limit point of {xi} ⊂ SE ⊂ SE∗∗ . Then by (α) we have that

G(h|E) = F0(h), for every h ∈ D, (3.5)

and by (β) we obtainG(f0|E) = F0(f0). (3.6)

By taking into account consequently (3.4), (3.5), (3.4), and (3.6), we conclude that

supG(K|E) = supG(co(D|E

)‖·‖= supG(D|E) =

supF0(D) ≤ supF0(C) = α < F0(f0) = G(f0|E),contradicting f0 ∈ K. The proof is complete. �

The following example shows that the FSP ofB plays a crucial role in Theorem 3.4, i.e.:if we substitute FSP by the weaker condition B|E is separable, for any separable subspaceE ⊂ X, then the conclusion B has (S) may not be true.

Example 3.5. Let X = C([0, ω1]

)with the supremum norm, K = BX∗ , and B ={

± δx : x ∈ [0, ω1)}. It is known that for any x ∈ X there is an ordinal α < ω1 such

that x restricted to [α, ω1] is constant. It follows that B is a boundary of K. Since X isan Asplund space it follows that B|E is separable, for any separable subspace E ⊂ X.

However, B has no (S). Indeed, for every measure µ ∈ co{± δx : x ∈ [0, ω1)

}we have

that indeed ‖δω1−µ‖ ≥ 1: notice that there exists a non-negative continuous function withvalues 0 at ω1 and 1 on the support of the measure µ.

Lemma 3.6. Let X be a Banach space. If there exists a boundary B ⊂ BX∗ which is FSP,then X is an Asplund space.


Proof. Let ξ be a map saying that B is FSP, see Definition 3.1. Let E ⊂ X be a closedseparable subspace and let D = {e1, e2, . . . , en, . . . } a dense subset of E. If we writeσn := {e1, e2, . . . , en} and D = ∪nξ(σn) then

B|E ⊂ co(D|E).

Hence B|E is a separable boundary for BE∗ . Thus E∗ is separable, [30, 14, 13], and X isAsplund. �

To prove our main Theorem 3.9 we need the following lemmata.

Lemma 3.7. Let X be a Banach space. If there exists a boundary B1 ⊂ BX∗ which isFSP, then any other boundary B ⊂ BX∗ with property (S) is FSP.

Proof. Let ξ1 be a map saying that B1 is FSP. We construct a mapping ξ in the followingway. Let σ be a finite subset of X and ξ1(σ) = {fi}mi=1. Using that B has property (S) wecan find a countable subset {hj} ⊂ B with ξ1(σ) ⊂ co{hj}. Let us define ξ(σ) = {hj}.We obviously have

ξ1(σ) ⊂ co ξ(σ). (3.7)

We claim that for any increasing sequence σn of finite subsets of X if we write E =[∪∞n=1 σn

]and D = ∪∞n=1ξ(σn), then we have

B|E ⊂ co(D|E). (3.8)

Indeed, first we note that since the boundary B1 is FSP, then B1|E is a separable boundaryof BE∗ and hence it has property (S), i.e.,

BE∗ = co(B1|E). (3.9)

Next by using (3.9), FSP property of B1 and (3.7) we obtain that

B|E ⊂ BE∗ = co(B1|E) ⊂ co(∪∞n=1ξ1(σn)|E) ⊂ co(D|E),

which proves (3.8). Now we can apply Proposition 3.3 to finish the proof. �

Lemma 3.8. Let f0 : X → SX∗ be a first Baire class map such that for any x ∈ X wehave f0(x)(x) = ||x||. Let fn : X → BX∗ , n = 1, 2, . . . , be a sequence of continuousmaps with limn fn(x) = f0(x), for any x ∈ X . If B1 := ∪∞n=0fn(X) then B1 is a FSPboundary.

Proof. For any finite subset σ of X put Eσ = [σ]. Let Aσ be a 1|σ|− net in SEσ . Put

ξ1(σ) =⋃x∈Aσ

{fn(x)}∞n=1 ⊂ B1.

Let σn be an increasing sequence of finite subsets of X , E =[∪∞n=1 σn

]and D =

∪∞n=1ξ1(σn). We claim thatf0(SE) ⊂ D (3.10)

Indeed, fix ε > 0 and x0 ∈ SE . Let m be such that ||f0(x0)− fm(x0)|| < ε/2. Since fmis continuous it follows that there is a δ > 0 such that for any x ∈ X with ||x − x0|| < δwe have ||fm(x) − fm(x0)|| < ε/2. Take n so large that 1/|σn| < δ/2 and that for somey ∈ SEσn we have ||y−x0|| < δ/2. Pick x ∈ Aσn with ||x−y|| < 1/|σn| < δ/2. An easycalculation shows that ||fm(x)− f0(x0)|| < ε and therefore our claim has been proved.


From (3.10) we obtain that f0(SE)|E is a separable boundary of BE∗ and consequentlyf0(SE)|E has property (S), i.e., co(f0(SE)|E) = BE∗ . Again from (3.10) follows that

B1|E ⊂ BE∗ = co(f0(SE)|E) ⊂ co(D|E).Now we can apply Lemma 3.3 to finish the proof. �

A wide class of FSP boundaries is provided by σ−fragmentable selectors of the dualitymapping J : X → 2BX∗ that sends each x ∈ X to the set

J(x) := {x∗ ∈ BX∗ : x∗(x) = ‖x‖},see Corollary 4.4 in Section 4.

Theorem 3.9. Let X be an Asplund space and B be a boundary of X. Then B has (S) ifand only if B is FSP.

Proof. If B is FSP then by Theorem 3.4 B has (S). Conversely we now prove that everystrong boundary is FSP. Since X is an Asplund space it follows that the duality mappingJ : X → 2BX∗ has a first Baire class selector

f0 : X → SX∗ ,

see for instance Theorem I.4.2 in [5]. The selector f0 has associated a sequence fn :X → BX∗ , n = 1, 2, . . . of continuous maps with limn fn(x) = f0(x), for any x ∈ X .The boundary B1 := ∪∞n=0fn(X) is a FSP boundary by Lemma 3.8 and Lemma 3.7 thatfinishes the proof. �

The following theorem gives a characterization of Asplund spaces in terms of bound-aries.

Theorem 3.10. The following statements for a Banach space X are equivalent:(i) X is an Asplund space;

(ii) X admits an FSP boundary;(iii) Any boundary B with (S) is FSP.

Proof. Implication (i)⇒(ii) follows from Theorem 3.9 for, say B = SX∗ . For the implica-tion (ii)⇒(iii) apply Lemma 3.7. Finally, to prove (iii)⇒(i) we take B = BX∗ ; clearly Bhas property (S), and by (iii) B is FSP, and (i) follows. The proof is complete. �

4. σ−FRAGMENTED MAPS

We shall deal in this section with boundaries constructed with σ-fragmentable maps.This class of maps is wide enough as to include all Borel measurable maps between com-plete metric spaces: σ-fragmentable maps were introduced in [17] and they have been ex-tensively studied in [20], [26] and [2]. Let us introduce them with the following property,see [2, Section 2] for a complete characterization:

Definition 4.1. A map f : T → E is σ-fragmented if, and only if, f is the pointwiselimit of a sequence of maps fn : T → E, such that for every n ∈ N there are sets{Tnm,m = 1, 2, · · · } with T = ∪∞m=1T

nm and such that for every m ∈ N and every closed

subset F ⊂ Tnm the restriction fn|F has at least a point of continuity,

An important property of σ-fragmented maps between metric spaces is that they sendseparable subsets of the domain space into separable subsets of the range space in theprecise way described in the theorem below:


Theorem 4.2. [26, Theorem 2.14] Let (T, d) and (E, ρ) be metric spaces and f : T → Ea σ-fragmented map. Then, for every t ∈ T there exists a countable set Wt ⊂ T such that

f(t) ∈⋃{f(Wtn) : n = 1, 2, . . .}

ρ

whenever {tn} is a sequence converging to t in (T, d). In particular, f(S) is separablewhenever S is a separable subset of T .

Whereas the above result has been used in [26] as an important tool for renorming inBanach spaces we will use it here as the key result to prove Theorem 4.3. We stress thatit has been known for a long time that Borel maps from a complete metric space into ametric space send separable subsets of the domain into separable subsets of the range,see for instance [33, Theorem 4.3.8]. It should be noted that σ-fragmented maps are notnecessarily Borel measurable though: for instance, every map between metric spaces withseparable range is σ-fragmented. Let us remark that a map with domain a metric spaceand with values in a normed space is Baire one if, and only if, it is σ-fragmented and thepreimage of open sets are Fσ sets, see [20, Chapter 2] and [15].

We can prove now a localized version of one of the main results in [2].

Theorem 4.3. Let X be a Banach space. Assume that K ⊂ X∗ is a w∗−compact convexset and f : X → K is a σ−fragmented selector for the attaining map

FK(x) := {k ∈ K : k(x) = sup {g(x) : g ∈ K}} .Then

co f(X)‖·‖

= K.

Proof. Let us prove that f(X) is an FSP boundary of K and the result here will followfrom Theorem 3.4.

By Theorem 4.2 there is a map φ from X into the family of all countable subsets off(X) such that

f(x) ∈ ∪nφ(xn),whenever x = limn xn, x, xn ∈ X .

For any finite subset σ of SX put Eσ = [σ] and select Aσ as a 1|σ|− net in the finite

dimensional sphere SEσ . Now we define the map

ψ(σ) = ∪x∈Aσφ(x) ⊂ f(X).

Let σn be an increasing sequence of finite subsets of X . Put E =[∪∞n=1 σn

]and D =

∪∞n=1ψ(σn). We will prove now that

f(SE) ⊂ D. (4.1)

Fix x0 ∈ SE . Since the sequence σn is increasing there are points xp ∈ SEσnp , p =

1, 2, . . . with n1 < n2 < . . . < np < . . . and limp ‖xp − x0‖ = 0. Let us choosefor every p an element yp ∈ Aσnp such that ‖xp − yp‖ ≤ 1

|σnp |. We consequently have

limp ‖x0 − yp‖ = 0 and thus f(x0) ∈ ∪∞p=1φ(yp), so f(x0) ∈ ∪∞n=1ψ(σn) and the prooffor (4.1) is finished.

By our hypothesis the set (f(SE))|E is a separable boundary of K|E . Since sepa-rable boundaries are strong, see Theorem I.2 in [14],[30] or [13], we have that K|E ⊂co(f(SE)|E , thus f(X)|E ⊂ co(D|E) which proves the FSP property of f(X) and fin-ishes the proof. �


Corollary 4.4. Let X be a Banach space and let f be a σ−fragmented selector for theduality mapping J : X → 2BX∗ . Then f(X) is a FSP boundary.

The following Corollary stresses Remark I. 5.1 in [5]

Corollary 4.5. Let X be a Banach space and J : X → 2BX∗ the duality mapping. Thefollowing statements are equivalent:

(i) X is an Asplund space.(ii) J has a Baire one selector.

(iii) J has a σ−fragmented selector.(iv) J has a selector f : X → X∗ such that f(X) is FSP.

Proof. For the implication (i)⇒(ii) we use that if X is Asplund, then Theorem 8 in [19]provides a Baire one selector for J , see also Theorem I.4.2 in [5]. The implication (ii)⇒(iii) follows from the fact that every Baire one map is σ-fragmentable, see Corollary 7in [17]. Finally, (iii) ⇒(iv) follows from Corollary 4.4. Finally, (iv) ⇒(i) follows fromTheorem 3.10. �

5. STRENGTHENING THE (I)-PROPERTY AND DESCRIPTIVE BOUNDARIES

Given x∗ ∈ X∗, D ⊂ X and ε > 0 we write

V (x∗, D, ε) := {y∗ ∈ X∗ : |y∗(x)− x∗(x)| ≤ ε, for every x ∈ D}.Denote by CBX the family of countable subsets of BX . Note that, while the family{

V (x∗, D, ε)}D∈FX ,ε>0

is a basis of w∗-neighborhoods for x∗, the family{V (x∗, D, ε)

}D∈CBX ,ε>0

is a basis of neighborhoods for x∗ for the locally convex topology γ in X∗ of uniformconvergence on bounded and countable subsets of X . The topology γ was used in [29]to characterize Asplund spaces X are those for which (X∗, γ) is Lindelof. Other paperswhere topology γ has been studied are [2, 3] and [4].

Recall that a topological space Y is Lindelof if every family of closed subsets of Y withempty intersection contains a countable subcollection with empty intersection.

We start with the next easy lemma.

Lemma 5.1. Let C be a w∗-compact (resp. γ-Lindelof) subset of X∗. For given z∗0 ∈ X∗and δ > 0 the following statements are equivalent:

(i) B[z∗0 , δ] ∩ C 6= ∅;(ii) V (z∗0 , D, δ) ∩ C 6= ∅ for each finite (resp. countable) set D ⊂ BX .

Proof. We prove the case C being γ-Lindelof. Since B[z∗0 , δ] ⊂ V (z∗0 , D, δ) it is clearthat (i)⇒( ii). Conversely, if we assume that (ii) holds then the family {V (z∗0 , D, δ) ∩ C :D ∈ CBX} is made up of γ-closed subsets of C with the property that for every countablesubfamily

{V (z∗0 , Dn, δ) ∩ C : n ∈ N, Dn ∈ CBX},has not empty intersection because

∅ 6= V (z∗0 ,∪nDn, δ) ∩ C ⊂⋂n

V (z∗0 , Dn, δ) ∩ C.


Since, C is γ-Lindelof we conclude that

∅ 6=⋂

D∈CBX

V (z∗0 , D, δ) ∩ C = B[z∗0 , δ] ∩ C,

and the proof of (ii)⇒ (i) is finished in this case.The proof for the equivalence (i)⇔ (ii) when C is w∗-compact is similar to the one we

have already given for the case γ-Lindelof keeping in mind now that in compact spacesevery family of closed subsets with the finite intersection property has a non empty inter-section. �

As a tool for our subsequent study we need to quote first the following result that havebeen established in [2].

Proposition 5.2 ([2, Theorem 5.4]). Let X be a Banach space. The following statementare equivalent:

(i) `1 6⊂ X;(ii) for every w∗-compact convex subset K of X∗ and any boundary B of K we have

K = coBγ;

(iii) for every w∗-compact subset K of X∗, coKw∗= coK

γ .

For Asplund spaces the following strong version of the (I)-formula holds.

Proposition 5.3. Let X be an Asplund space, K a w∗−compact convex subset of the dualspace X∗ and B ⊂ K a boundary of K. Then, for any increasing sequence {Bn}∞n=1 ofsubsets of B with B = ∪nBn we have

K = ∪∞n=1coBnγ ||.||

. (SI)

Proof. Set B′ := ∪∞n=1coBnγ . B′ is a convex boundary of K. Thus, Proposition 5.2

applies to yield K = B′γ. On the other hand, since X is Asplund the space (X, γ) is

Lindelof, see [29] and [4]. ThereforeB′ is γ-Lindelof too and a straightforward applicationof Lemma 5.1 gives us B′

γ= B′ that combined with the equality K = B′

γfinishes the

proof. �

The same ideas that we have used in the previous proposition are used in the next onethat extends [27, Theorem 2.3].

Proposition 5.4. Let X be an Asplund space, K a w∗−compact convex subset of the dualspace X∗ and B ⊂ K a boundary of K. If B is γ-closed, then B is strong.

Proof. IfB is γ-closed, then for any n ∈ N the setBn is a closed subset of (X∗, γ)n that isγ-Lindelof, see [29] or [4, Theorem 2.3]; thus (B, γ|B)n is Lindelof. Now observe that if(B, γ|B)n is Lindelof for every n ∈ N then the convex hull coB is γ-Lindelof too. Indeed,we notice first that coB = ∪n conB where for every n ∈ N we have written

conB :={∑n

k=1 λkbk : 0 ≤ λk ≤ 1, bk ∈ B, k = 1, 2, . . . , n,∑n

k=1 λk = 1}

If Kn := {(λ1, λ2, . . . , λn) ∈ [0, 1]n :∑n

k=1 λk = 1}, then Kn is compact with thetopology induced by the product topology of [0, 1]n and therefore Kn × (B, γ|B)n is aLindelof space, [7, Corollary 3.8.10]. All things considered the map

ψn : Kn × (B, γ|B)n → (X∗, γ)


((λk)

nk , (bk)

nk

)→

n∑k=1

λkbk

is continuous and its image conB is therefore γ-Lindelof. Hence coB = ∪n conB is aγ-Lindelof convex boundary of K and we finally conclude that

KProp.5.2

= coBγ Lem.5.1

= coB.

The proof is over. �

A topological space (T, τ) is said to be K-analytic if there is an upper semi-continuousset-valued map F : NN → 2T such that F (σ) is compact for each σ ∈ NN and F (NN) :=⋃{F (σ) : σ ∈ NN} = T . Our basic reference for K-analytic spaces is [18].

We use the following conventions: N(N) is the set of finite sequences of positive integers;if α = (n1, n2, . . . , nk, . . . ) ∈ NN and if k ∈ N, then we write α|k := (n1, n2, ..., nk) ∈N(N). Notice that if α = (n1, n2, . . . , nk, . . . ) ∈ NN and for every k ∈ N we let

[n1, n2, . . . , nk] = {β ∈ NN : β|k = (n1, n2, . . . , nk)}then

([n1, n2, . . . , nk]

)∞k=1

is a basis of neighborhoods for α in NN.

Proposition 5.5. Let X be a Banach space and B : NN → 2X∗

a w∗-upper semicontinu-ous map such that Bα := B(α) is w∗-compact for every α ∈ NN. Then we have⋃

α∈NN

coBαw∗

=⋃α∈NN

coBαw∗γ

. (5.1)

Proof. Since the set in left hand side of (5.1) is clearly contained in set in the right handside, to prove the equality (5.1) we only have to prove that if

z∗0 6∈⋃α∈NN

coBαw∗

thenz∗0 6∈

⋃α∈NN

coBαw∗γ

.

Fix δ > 0 such that

B[z∗0 , δ] ∩ coBαw∗

= ∅, for every α ∈ NN.

We apply Lemma 5.1 for each w∗-compact set coBαw∗

to obtain a finite subset Dα of BXsuch that V (z∗0 , Dα, δ)∩ coBα

w∗= ∅. The separation theorem in (X∗, w∗) applied to the

w∗-closed set V (z∗0 , Dα, δ) and the w∗-compact set coBαw∗

provides us with xα ∈ X ,and λα > ξα in R such that

V (z∗0 , Dα, δ) ⊂ Gλα := {y∗ ∈ X∗ : xα(y∗) > λα} (5.2)

andcoBα

w∗⊂ Gξα := {y∗ ∈ X∗ : ξα > xα(y

∗)}.Since Gξα is w∗-open and Bα ⊂ Gξα , the w∗-upper semicontinuity of B implies that forsome kα ∈ N if we write α|kα = (n1, n2, . . . , nkα) we have that

B([n1, n2, . . . , nkα ]

):=⋃{B(β) : β ∈ NN, β|kα = α|kα}

)⊂

⊂ Gξα := {y∗ ∈ X∗ : ξα > xα(y∗)}


We notice that

coB([n1, n2, . . . , nkα ]

)w∗

⊂ {y∗ ∈ X∗ : ξα ≥ xα(y∗)}and since λα > ξα the inclusion (5.2) leads us to

V (z∗0 , Dα, δ) ∩ coB([n1, n2, . . . , nkα ]

)w∗

= ∅, for every α ∈ NN. (5.3)

Since N(N) is countable, the family

C := {B([n1, n2, . . . , nkα ]

): α ∈ NN}

is countable too and it can be written as C = {Dn : n ∈ N}. Now we can rewrite (5.3)in terms of the Dn’s in the following way: for every n ∈ N there is a finite set Fn ⊂ BXsuch that

V (z∗0 , Fn, δ) ∩ coDnw∗

= ∅.The latter implies that

V (z∗0 ,∪∞n=1Fn, δ) ∩[∪∞n=1 coDn

w∗]= ∅,

that implies that

z∗0 6∈⋃n∈N

coDnw∗γ

. (5.4)

We notice now that for every α ∈ NN we have Bα ⊂ B([n1, n2, . . . , nkα ]), and there-

fore (5.4) implies that z∗0 6∈⋃α∈NN coBα

w∗γ

, and the proof is over. �

We reach now a main result for us that extends [16, Theorem 3.3].

Theorem 5.6. Let X be a Banach space. The following statements are equivalent:(i) X does not contain `1;

(ii) for every w∗-compact subset K of X∗ any w∗-K-analytic boundary B of K isstrong;

(iii) for every w∗-compact subset C of X∗ we have

coCw∗

= coC. (5.5)

Proof. The equivalence (i)⇔ (iii) is [16, Theorem 3.3]: we explicitly keep this equivalencehere for further use. Since, every w∗-compact set C is w∗-K-analytic the implication (ii)⇒ (iii) is easily obtained when bearing in mind that C is a boundary of K := coC

w∗.

To finish we prove (i) ⇒ (ii). Let B be a w∗-K-analytic boundary of K and let T :NN → 2B a w∗-compact valued upper semicontinuous map such that B = ∪{T (α) : α ∈NN}. For each α ∈ NN the set {β ∈ N : β ≤ α} is compact in NN and therefore its image

B(α) := T({β ∈ N : β ≤ α}

)={T (β) : β ∈ NN, β ≤ α

}is w∗-compact. It is easily seen that B is w∗-upper semicontinuous and since T (α) ⊂B(α), for every α ∈ NN we obtain that B = ∪{B(α) : α ∈ NN}. The definition of Bclearly implies that B(α) ⊂ B(β) whenever α ≤ β. Observe that coB =

⋃α∈NN coB(α).

This allows us to finally obtain

K ⊃ coB =⋃α∈NN

coB(α)(iii)=

⋃α∈NN

coB(α)w∗ Prop.5.5

=

=⋃α∈NN

coB(α)w∗γ

⊃ coBγ Prop.5.2

= K.


The proof is finished. �

We stress that Godefroy proved in [14, Theorem III.3] that if K ⊂ X∗ is w∗-compactset and B is a weak-K-analytic boundary then B is strong. Notice that in general thehypothesis of w∗-K-analyticity is weaker than this of weak-K-analyticity: indeed, forevery non Asplund space X the dual unit ball BX∗ is w∗-K-analytic but it is not weakly-K-analytic: indeed, if BX∗ were weakly K-analytic then BX∗ would be weakly Lindelof,that is, X∗ would be weakly Lindelof and [6, Proposition 1.8] applies to conclude X isAsplund.

We need the following lemma in the proof of Proposition 5.8. Although the lemmaeasily follows from known duality arguments we include a proof to help with the readingof subsequent results.

Lemma 5.7. Let X be a Banach space, Y ⊂ X a subspace and let w∗ denote the weak∗

topology in X∗∗. The following properties hold:

(i) BYw∗

= BX∗∗ ∩ Y w∗;

(ii) if Y is separable, `1 6⊂ Y and D ⊂ Y is bounded, then(Dw∗, w∗

)is Rosenthal

compact.

Proof. Property i) follows from duality arguments in the dual pair 〈X∗∗, X∗〉. ConsiderBY = BX ∩ Y as a subset X∗∗. The bipolar theorem [25, §20.8.(5)] and the formulas forthe polar of an intersection and a union, [25, §20.8.(8) and §20.8.(9)] allow us to write

BYw∗

= (BY )◦◦ = [(BX ∩ Y )◦]◦ =

[aco

((BX)◦ ∪ Y ◦

)]◦=

= (BX)◦◦ ∩ Y ◦◦ = BX∗∗ ∩ Y w∗

.

Let us prove (ii). Write i : Y ↪→ X to denote the inclusion map from Y into X . Thebi-adjoint map i∗∗ : Y ∗∗ → X∗∗ is injective and w∗Y ∗∗-to-w∗ continuous. Thus we have

D ⊂ i∗∗(Dw∗Y ∗∗ ) ⊂ Dw∗

.

Since i∗∗(Dw∗Y ∗∗ ) is w∗-compact we obtain that

i∗∗(Dw∗Y ∗∗ ) = D

w∗

and therefore i∗∗ : (Dw∗Y ∗∗ , w∗Y ∗∗)→ (D

w∗, w∗) is an homeomorphism. If Y is separable

and `1 6⊂ Y then (Dw∗Y ∗∗ , w∗Y ∗∗) is Rosenthal compact, see [28], and therefore (D

w∗, w∗)

is Rosenthal compact too and the proof of (ii) is finished. �

Proposition 5.8. Let X be a Banach space not containing `1. The following propertieshold:

(i) For every countable and bounded set D ⊂ X its w∗-closure Dw∗in X∗∗ is con-

tained in (X∗, γ)′ and it is γ-equicontinuous. Furthermore,(Dw∗, w∗

)is Rosen-

thal compact.(ii) (X∗, γ)′ = {x∗∗ ∈ X∗∗ : limn xn = x∗∗ for some (xn)n ⊂ X}, where the limits

involved are taken in the w∗-topology.(iii) For every norm bounded countable set D′ ⊂ (X∗, γ)′ there is a bounded and

countable set D ⊂ X such that D′w∗⊂ Dw∗

. Hence, D′w∗

is a γ-equicontinuoussubset of (X∗, γ)′ and

(D′

w∗, w∗

)is Rosenthal compact.


Proof. The proof of (i) is as follows. If we consider D as a family of functionals definedon X∗ then D ⊂ X∗∗ is γ-equicontinuous. Therefore its pointwise closure in RX∗

, thatcoincides with its w∗-closure in X∗∗, denoted by Dw∗

, is γ-equicontinuous again and thusDw∗⊂ (X∗, γ)′ ⊂ X∗∗. Since Y := [D] is separable and `1 6⊂ X , we conclude that

`1 6⊂ Y and therefore statement (ii) in Lemma 5.7 implies that(Dw∗, w∗

)is Rosenthal

compact and the proof of (i) is finished.Statement (ii) easily follows from statement (i). Let us write

W := {x∗∗ ∈ X∗∗ : w∗ − limnxn = x∗∗ for some (xn)n ⊂ X}.

Note that the sequences (xn)n involved in the definition of W are necessarily bounded forthe norm. Therefore such a sequence (xn)n ⊂ (X∗, γ)′ is γ-equicontinuous and its limitx∗∗ = w∗ − limn xn is γ-continuous. This explain the inclusion W ⊂ (X∗, γ)′. The otherway around: we prove now that (X∗, γ)′ ⊂ W . If x∗∗ ∈ X∗∗ is γ-continuous, then thethere exist a norm bounded and countable subset D ⊂ X such that

|x∗∗(x∗)| < 1 for each x∗ ∈ V (0, D, 1).

In other words x∗∗ belongs to the absolute bipolar D◦◦ of D in (X∗, γ)′. The separation(Bipolar) theorem, [25, §20.8.(5)], implies that x∗∗ ∈ acoD

w∗, where acoD stands for

the absolutely convex hull of D. Hence x∗∗ ∈ acoQDw∗

. Since acoQD ⊂ X is countableand bounded, statement (i) applies to tell us that the space

(acoQD

w∗, w∗) is Rosenthal

compact, and in particular it is an angelic space, see [1]. Thus x∗∗ is the w∗-limit of asequence in acoQD. The latter says that (X∗, γ)′ ⊂W and the proof for (ii) is finished.

We prove (iii). Take D′ ⊂ (X∗, γ)′ countable and norm bounded. We can and doassume that D′ ⊂ BX∗∗ . Statement (ii) ensures us of the existence of F ⊂ X countablesuch that D′ ⊂ Fw

∗–we do not assume that F is bounded. If we write Y := [F ] then Y is

separable and statement (i) in Lemma 5.7 says that BYw∗

= BX∗∗ ∩ Y w∗. Take D ⊂ BY

countable and norm dense: the inclusionD′ ⊂ BYw∗

implies thatD′ ⊂ Dw∗and the proof

of the first part of (iii) is finished. Once this is done statement (i) applied to Dw∗gives us

the second part of (iii) and the proof is completed. �

Our previous work allows us to prove that the sup − lim sup property (see the equality(SLS) in the introduction) can be extended to more general functions. This result appearsas one of the ultimate forms of the vintage Rainwater’s theorem on sequential convergenceon extreme points, see Theorem 3.60 in [9].

Theorem 5.9. Let X be a Banach space not containing `1, K a w∗−compact convexsubset of the dual space X∗ and B ⊂ K a boundary of K. Let (zn)n be a boundedsequence in (X∗, γ)′ then

supb∗∈B{lim sup

n→∞zn(b

∗)} = supx∗∈K

{lim supn→∞

zn(x∗)}. (5.6)

Proof. Write l for the left hand side in (5.6). Since for any two sequences in R we havelim sup(sn + tn) ≤ lim sup sn + lim sup tn , we easily obtain that

l = supb∗∈coB

{lim supn→∞

zn(b∗)}. (5.7)

On one hand (iii) in Proposition 5.8 says that the set {zn : n ∈ N} is a γ-equicontinuoussubset of (X∗, γ)′. On the other hand, Proposition 5.2 gives us the equality K = coB

γ .Fix ε > 0. Given any x∗ ∈ K there is b∗ ∈ coB such that |zn(x∗) − zn(b∗)| < ε for


every n ∈ N. The latter together with (5.7) imply that lim supn→∞ zn(x∗) ≤ l + ε. Since

x∗ ∈ K is arbitrary we conclude that

supx∗∈K

{lim supn→∞

zn(x∗)} ≤ l + ε,

for every ε. Hence equality (5.6) holds and the proof is over. �

Remark 5.10. Note that if the Banach spaceX has the property that for everyw∗−compactconvex subset of the dual space X∗ and any boundary B of K the thesis of Proposition 5.9holds, thenX does not contain `1. Indeed, equality (5.6) implies in particular that for everyx∗∗ ∈ (X∗, γ)′ we have

supx∗∈B

x∗∗(x∗) = supx∗∈K

x∗∗(x∗).

The latter says that K = coBγ and now Proposition 5.2 applies to give us that X cannot

contain `1. �

We finish the paper with the following question that appears in [14, Question V.1] thatseems to be still open.

Question 5.11. Let X be a separable Banach space with `1 6⊂ X and E the set of w∗-exposed points of BX∗ . Is it true that BX∗ = co E‖·‖.

Acknowledgements.- We gratefully thank the referee for the comments intended to im-prove this paper.

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Boundaries of Asplund spaces

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